Some animals do sweat because they have the pores to do so such as cattle and horses.

After a long ride you take off a saddle and there will be sweat.
Dogs & cats for instance cannot sweat so they pant to relieve themselves of heat. Some animals have different ways to relieve the build up as their bodies take care of it in different ways.
I’ve always found it interesting that a polar bears skin is black to keep them warm and the fur is white to let them blend and when they still get to cold they will hide out under the snow using the packing to get away from the worst of the cold.

Not all animals sweat. I remember being told as a child that only humans and horses sweat. This isn’t exactly true, either. Several different species of animals sweat, but more don’t than do. First of all, only warm-blooded animals have a need to sweat. This means we can immediately eliminate all fish, reptiles, amphibians, worms, insects, and arachnids. These are all cold-blooded animals. Cold-blooded animals don’t have a constant body temperature. Their body temperature changes with the environment, so they don’t really get hot or cold like we do.

Warm-blooded animals include mammals and birds. Warm-blooded animals maintain a constant body temperature. When they get too hot, they have to cool down. When they get too cold, they have to expend energy to keep them warm. Sweating is one of the ways warm-blooded animals keep cool, however, only mammals sweat. Birds keep cool by using shade, bristling out their feathers, fluttering their throats, and panting. Many mammals, such as dogs, also use panting to keep cool. Panting is form of breathing that brings in extra air to cool the body and take out extra heat from inside the body.

Many mammals sweat, but none as profusely as humans. Humans have sweat glands all over their bodies. Dogs and cats sweat, but they only sweat from the pads of their feet. Many other four-legged mammals follow this foot-only sweating pattern. Horses, as mentioned, sweat from most areas of their bodies, except their hooves. Pigs don’t sweat at all. That’s why they roll around in the dirt and mud. It cools them off and keeps direct sunlight off their skin. Cats keep cool by licking themselves. The moisture from their tongues acts like sweat does on our skin. If they are extremely hot, cats will pant like dogs.